Optimal design of superstructures for placing units and streams with multiple and ordered available locations. Part I: A new mathematical framework

Highlights

• A decomposition strategy is used to optimize superstructures.

• Optimal placement of streams and process units over a superstructure.

• Optimization of superstructures expressed as Mixed-Integer Nonlinear Problems.

• A time-efficient Discrete-Steepest Descent Algorithm (D-SDA) is presented.

• A CSTR network is used to illustrate the implementation of the proposed algorithm.

Abstract

A new approach for the optimal design of superstructures in chemical engineering is proposed in this study. Contrary to most of the optimization techniques established in the literature, this approximation exploits the structure of a specific type of problem, i.e., the case where it is necessary to find the optimal location of a processing unit or a stream over a naturally ordered discrete set. The proposed methodology consists of reformulating the binary variables of the original Mixed-Integer Nonlinear Problem (MINLP) with a smaller set of integer variables referred to as external variables. Then, the reformulated optimization problem can be decomposed into a master Integer Program with Linear Constraints (master IPLC) and primal sub-problems in the form of Fixed Nonlinear Programs (FNLPs), i.e., Nonlinear Programs (NLPs) with integer variables fixed. The use of the Discrete-Steepest Descent Algorithm (D-SDA) is considered for the master IPLC, while the primal FNLPs are solved with existing Nonlinear Programming (NLP) solvers. The main features of this approach are discussed with an illustrative example: an isothermal Continuously Stirred Tank Reactor (CSTR) network with recycle and autocatalytic reaction. The new methodology does not guarantee global optimality; however, the results show that it can find a local solution in a short computational time.

Introduction

A relevant difficulty when optimizing the configuration and operating conditions of a superstructure representing a chemical process is the nonlinear nature of its mathematical model. The optimal design of a chemical process typically results in a Mixed-Integer Nonlinear Program (MINLP), where the nonlinearities may arise from complicated expressions in the continuous domain, e.g., equilibrium relationships and reaction rates. Additionally, the presence of integer variables complicates the optimization due to the nonlinear expressions in the integer-continuous domain and the nonlinear relations in the integer domain only (Floudas, 1995). These nonlinearities result in optimization problems where acceptable local solutions are difficult to obtain because global optimality conditions cannot be guaranteed (Yeomans and Grossmann, 2000). Under these circumstances, general-purpose solvers may not be able to find local solutions; however, knowledge of special problem structures and systematic characteristics can lead to the development of robust and specialized algorithms (Chen and Grossmann, 2017).

The exploitation of the problem structure can yield to simplifications in the step of modeling and improvements in terms of algorithm efficiency and robustness (Grossmann et al., 1999). For example, the Generalized Disjunctive Programming (GDP) formulation serves as a tool to represent the logic behind process networks (Grossmann and Ruiz, 2012). Besides, the GDP representation has encouraged the use of logic-based optimization methods like the Logic-Based Outer Approximation and the Logic-Based Benders Decomposition (Türkay and Grossmann, 1996). This framework, i.e., the GDP formulation and the logic-based method, allowed to disregard unnecessary constraints during the solution procedure and to avoid zero-flow singularities. Similarly, the objective of this series of articles (Parts I and II) is to provide a new mathematical framework that exploits the specific structure of a persistent class of problems in the field of optimization-based process synthesis.

A typical challenge that arises when optimizing superstructures is the optimal placement of streams and process units. An example of optimal placement of streams is the optimal reflux and feed location in separation-reaction networks. Similarly, the optimal number of consecutive or parallel process units can be mathematically modeled in terms of the position of the last unit over a grid. The common factor of these examples is that the available locations can be arranged in the order defined by the natural numbers, subsequently, the optimization problem can be expressed in terms of variables that represent positions over a one-dimensional discrete space. This representation is referred to as the external variables reformulation. To the best of our knowledge, this is the first study that presents the use of the external variables reformulation for superstructure optimization. Networks for synthesizing a processing system are typically modeled using binary, Boolean and Special Ordered Sets (SOS) of variables through the MINLP and the GDP formulations (Garcia and You, 2015; Grossmann and Ruiz, 2012). Therefore, a reformulation is required to transform an MINLP with binary variables into an optimization problem with external variables. The resulting problem is still an MINLP, where some sets of binary variables are expressed in terms of a smaller set of integer variables.

Another contribution of this study is the development of a deterministic method for the mixed-integer optimal design of superstructures reformulated with external variables. The method is designed to find local solutions efficiently, but those solutions are not guaranteed to be globally optimal. The proposed method overcomes some limitations caused by nonlinearities; specifically, the nonlinearities in the integer and integer-continuous domains generated by the binary variables that can be reformulated with external variables. These difficulties are successfully handled by decomposing the problem into a master problem and primal sub-problems. The master problem is constructed from the external variables set, and it finds feasible solutions over the external variables’ search space. The primal sub-problems are obtained by fixing the binary terms that can be expressed in terms of the external variables. This strategy is attractive due to the avoidance of complicating variables (e.g., integer variables) in the primal sub-problems, as it also occurs with the Outer Approximation and the Benders decomposition (Duran and Grossmann, 1986; Nowak, 2005; Rahmaniani et al., 2017).

The proposed reformulation and decomposition are useful to apply an optimization algorithm for the master problem that guarantees fixed integer values for the external variables in the primal sub-problems. An algorithm that is well-suited to this situation is the Discrete-Steepest Descent Algorithm (D-SDA) proposed by Murota (2003). This algorithm is based on discrete convex analysis theory, which aims to establish a theoretical framework for solving discrete optimization problems by combining continuous and combinatorial optimization theory. The theory of continuous convex analysis can be extended to discrete settings without considering the usual definition of convexity in Integer Programs, i.e., if a convex problem is obtained when relaxing the integer requirement, then the Integer Program is convex (Kronqvist et al., 2019). Instead, the field of discrete convex analysis uses its own definition, i.e., integral convexity. Subsequently, the optimality criterion of any discrete convex analysis-based method is different from those used by the available Integer Programming solvers. This result is useful, because a discrete problem may be integrally convex, even if the problem is non-convex according to the usual definition of convexity. Then, if a discrete convex analysis-based method is used to solve a problem, different and maybe better local solutions can be found. This is a motivation to implement the D-SDA for the optimization of superstructures, where the challenge of this study relies on applying this Integer Programming algorithm to an MINLP problem reformulated with external variables.

During the last two decades, discrete convex analysis has been recognized as a powerful framework for analyzing discrete optimization problems (Murota, 2016). Favati and Tardella (1990) were the first to introduce the concept of integrally convex functions, and to propose a local characterization for global optimality. Afterward, Murota (2000) proposed the D-SDA to optimize functions defined on integer lattices. This algorithm was improved by Iwata (2002), with the implementation of a scaling technique to speed up the algorithm. This theory has been used as a tool in economics, game theory and network flow problems (Murota, 2016); nevertheless, applications of this approach in chemical engineering have not been reported. As it will be shown in this study, the external variables representation of the MINLP problem; the decomposition in master and primal sub-problems, and the discrete convex analysis theory allow the D-SDA to be applied for the optimal design of superstructures that often emerge in chemical engineering applications.

This series of studies (Parts I and II) considers reformulated MINLP problems with external variables, where: (1) every set of binary variables can be expressed in terms of a smaller set of variables, i.e., the external variables; (2) every linear logic constraint can be expressed in the external variables’ domain, and (3) the integer constraints in the external variables’ domain are linear, i.e., the external variables are restricted over a convex polyhedron. This results in a master Integer Program with Linear Constraints (master IPLC) that optimizes the external variables and primal sub-problems with discrete variables fixed at integer values. These sub-problems are called Fixed Nonlinear Programs (FNLPs), where the binary terms are treated as parameters calculated from the external variables’ values. The main contributions of the first part of this series of articles are summarized as follows:

• A new modeling framework for process synthesis is proposed. In contrast to the existing MINLP strategies, this approach is specifically designed to allocate process units and streams.

• A specialized algorithm to solve a problem reformulated with the proposed mathematical framework is developed.

The general mathematical framework for reformulating and decomposing an MINLP superstructure for placing units and streams with multiple and ordered available locations is presented at the beginning of Section 2. Subsequently, the optimality conditions based on discrete convex analysis are described at the end of Section 2. These optimality conditions allow the D-SDA to be adapted for solving MINLP superstructures that have been reformulated with external variables and decomposed into a master IPLC and primal FNLPs (see Section 3). The main features of the proposed methodology are illustrated through the optimization of a Continuously Stirred Tank Reactor (CSTR) network in Section 4. Finally, the conclusions and future work are presented in Section 5.